Online Encyclopedia
List of publications in mathematics
This is a list of important publications in mathematics, organized by field.
Some reasons why a particular publication might be regarded as important:
- Topic creator - A publication that created a new topic
- Breakthrough - A publication that changed scientific knowledge significantly
- Introduction - A publication that is a good introduction or survey of a topic
- Impact - A publication which had a major impact on the world or on the research
- Latest and greatest - The current most advanced result in a topic
Geometry
- Author: Euclid
- Publication data: c. 300 BC
- Online version: Interactive Javascript version
- Description: This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The Elements is still a valuable resource and a good introduction to algorithm. More than any specific result in the publication, it seems that the major achievement of this publication is the popularization of logic and mathematical proof as a method of solving problems.
- Importance: Topic creator, Breakthrough, Impact, Introduction, Latest and greatest (though it is the first, some of the results are still the latest)
- Author: René Descartes
- Publication data:
- Online version:
- Description: La Géométrie was published in 1637 and written by René Descartes. The book was influential in developing the Cartesian coordinate system and specifically discussed the representation of points of a plane, via real numbers; and the representation of curves, via equations.
- Importance: Topic creator, Breakthrough, Impact
Géométrie Algébrique et Géométrie Analytique
- Author: Jean-Pierre Serre
- Publication data:
- Online version:
- Description: In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geomety to include, for example, techniques from Hodge theory. (NB While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. A GAGA-style result would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings.
- Importance: Topic creator, Breakthrough, Impact
Logic
Principia Mathematica
Description: The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Bertrand Russell and Alfred North Whitehead and published in 1910-1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled, in a rather disappointing way, by Gödel's incompleteness theorem in 1931.
Importance: Impact
Gödel's incompleteness theorem
- Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931).
- Kurt Gödel
- Online version
Description: In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1930. Somewhat simplified, the first theorem states:
In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system.
Importance: Breakthrough ,Impact
Information theory
See the list of publications in information theory.
Number theory
Disquisitiones Arithmeticae
Description: The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own.
Importance: Breakthrough , Impact
On the Number of Primes Less Than a Given Magnitude
Description: On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. Importance: Breakthrough , Impact
Vorlesungen über Zahlentheorie
Description: Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory
Importance: Breakthrough , Impact
Calculus
Philosophiae Naturalis Principia Mathematica
Description: The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687. Probably the most influential scientific book ever published, it contains the statement of Newton's laws of motion forming the foundation of classical mechanics as well as his law of universal gravitation. He derives Kepler's laws for the motion of the planets (which were first obtained empirically). In formulating his physical theories, Newton had developed a field of mathematics known as calculus.
Up to the publication of this book, mathematics was only used to decribe nature. This is the first instance when mathematics is used to explain nature. Here was born the practice, now so standard we identify it with science, of explaining nature by postulating mathematical axioms and demonstrating that their conclusion are observable phenomena. In other words, the greatness of the Principia is not only in developing a number of fundamental thoeries in physics and mathematics but first and foremost (amply demonstrated in the title!) in the very linking of science and mathematics. The influence of this book is so deep that nowadays we find this link obvious and cannot imagine doing science in any other way.
Importance: Topic creator, Breakthrough , Impact
Numerical analysis
Method of Fluxions
Description: Method of Fluxions was a book written by Isaac Newton. The book was completed in 1671, and published in 1736.
Within this book, Newton describes a method (the Newton-Raphson method) for finding the real zeroes of a function.
Importance: Topic creator, Breakthrough , Impact
Game theory
The Theory of Games and Economic Behavior
- Oskar Morgenstern, John von Neumann
- Oskar Morgenstern, John von Neumann: The Theory of Games and Economic Behavior, 3rd ed., Princeton University Press 1953
Description: This book led to the investigation of modern game theory as a prominent branch of mathematics. This profound work contained the method -- alluded to above -- for finding optimal solutions for two-person zero-sum games.
Importance: Impact, Topic creator, Breakthrough
On Numbers and Games
Description: The book is in two, {0,1|}, parts. The zeroth part is about numbers, the first part about games - both the values of games and also some real games that can be played such as Nim, Hackenbush , Col and Snort amongst the many described.
Importance:
Winning Ways for your Mathematical Plays
- Elwyn Berlekamp, John Conway and Richard Guy
Description: A compendium of information on mathematical games. It was first published in 1982 in two volumes, one focusing on Conway's game theory and surreal numbers, and the other concentrating on a number of specific games. Importance:
Fractals
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
Description: A discussion of self-similar curves that have fractional dimensions between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975.
Importance: Shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.
Early manuscripts
These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the History of mathematics.
Rhind Papyrus
Description: It is one of the oldest mathematical texts, dating to somewhere around 17th century BC. Written by the scribe Ahmes or Ah'mose. Besides describing how to obtain an approximation of π only missing the mark by under one per cent, it is describes one of the earliest attempts at squaring the circle and in the process provides persuasive evidence against the theory that the Egyptians deliberately built their pyramids to enshrine the value of π in the proportions. Even though it would be a strong overstatement to suggest that the papyrus represents even rudimentary attempts at analytical geometry, Ahmes did make use of a kind of an analogue of the cotangent.
The Nine Chapters on the Mathematical Art
- unknown author
Description: a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. Among its content: Linear problems solved using the principle known later in the West as the rule of false position. Problems with several unknowns, solved by a principle similar to Gaussian elimination. Problems involving the principle known in the West as the Pythagorean theorem.
Archimedes Palimpsest
Description: Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry, he used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a parabola and one of its secant lines. Contrary to historically ignorant statements found in some 20th-century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. For explicit details of the method used, see how Archimedes used infinitesimals.
The Sand Reckoner
Description: The first known (European) system of number-naming that can be expanded beyond the needs of everyday life.
Textbooks
Course of Pure Mathematics
Description: A classic textbook in introductory mathematical analysis, written by G. H. Hardy. It was first published in 1908, and went through many editions. It was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge, and in schools preparing pupils to study mathematics at Cambridge. As such, it was aimed directly at "scholarship level" students — the top 10% to 20% by ability. The book contains a large number of difficult problems. The content covers introductory calculus and the theory of infinite series.
Art of Problem Solving
- Richard Rusczyk and Sandor Lehoczky
Description: The Art of Problem Solving began as a set of two books coauthored by Richard Rusczyk and Sandor Lehoczky. The books, which are about 750 pages together, are for students who are interested in math and/or compete in math competitions .
Popular writing
Gödel, Escher, Bach
Description: Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book, first published in 1979 by Basic Books. It is a book about how the creative achievements of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach interweave. As the author states: "I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."